The root-class residuality of the fundamental groups of graphs of groups

Consider a class $\mathcal C$ of groups containing at least one nontrivial group and closed under subgroups, extensions, and Cartesian products of the form $\prod\nolimits_{y \in Y}X_{y}$, where $X, Y \in {\mathcal C}$ and $X_{y}$ is an isomorphic copy of $X$ for each $y \in Y$. Given an arbitrary graph of groups, we obtain necessary and sufficient conditions for its fundamental group $G$ to be a residually $\mathcal C$-group thus generalizing Shirvani's conditions for the residual finiteness of $G$.